What is causing the bright edges on this mesh when using volume emission shaders?
Your mesh apears to be non-manifold. Specifically, it looks like you have split faces, or 'holes'. Volumetric shaders require manifold geometry to work correctly. Remove doubles, ensure that the geometry is clean, and do another test render. 

Edit: Apply Scale Ctrl A > S should also be done for this object. Note that your object has a scale of 0.006 which can have consequences and cause artifacts. Try to keep a scale of 1 on your object, unless there is a specific reason not to. See Campbell's explaination on why/how non 1:1:1 scaling and rotations can cause issues within Blender (This is true for other 3D apps as well).